As the advancement of hardware, image level of images handled by peripheral devices, such as a computer and a digital camera/printer, now matches that handled by an ultraprecise analog camera. Naturally, the data amount contained in an image increases. Therefore, when display speed is considered more important than precise image detail, an image compression is a requisite technique. Ideally, this image compression not only reduces the data amount, but compresses an original image to have the difference therefrom unrecognizable by person's eyesight when it is decompressed. At the same time, if a size of a storage area can be smaller, it is possible to effectively use computer resources, such as a memory and a hard disk.
At Web sites, images compressed in a format of such as GIF, JPEG and PNG are generally used. These compressed images are appropriate for Web sites which require a high accessibility since a file size thereof is significantly suppressed when compared to that of an uncompressed image, such as a BMP image. In addition, an image compression technique is often highly appreciated when image data is transmitted and received via a network.
Light and dark shading of color of an image is digitalized inside a computer for each pixel. For example, 256 tone gray scale image X having m number of vertical pixels and n number of horizontal pixels is represented as m×n matrix having integer values of [0, 255] as its components.
                              X          =                      (                                                                                x                    11                                                                                        x                    12                                                                    …                                                                      x                                          1                      ,                      n                                                                                                                                        x                    21                                                                                        x                    22                                                                    …                                                                      x                                          2                      ,                      n                                                                                                                    ⋮                                                  ⋮                                                  ⋱                                                  ⋮                                                                                                  x                                          m                      ,                      1                                                                                                            x                                          m                      ,                      2                                                                                        …                                                                      x                                          m                      ,                      n                                                                                            )                          ,                            [                  Expression          ⁢                                          ⁢          1                ]            where xi, j are integer values of [0, 255].
Hereinafter, an image and a matrix are represented as identical. An RGB color image holds light and dark shading information regarding three colors of Red, Green and Blue, respectively, in a similar matrix form to that of a gray scale image. Accordingly, for simplicity, the gray scale image will be described hereinafter.
A two-dimensional discrete Wavelet transformation or a blocking+singular value decomposition is performed once on image X. Image X1, which is like a one-half size of image X, is generated at an upper-left corner of (m/2)×(m/2) pixels. Edge extracted images in a vertical direction, a horizontal direction and a diagonal direction appear at upper-right corner, a lower-left corner and at a lower-right corner, respectively. Similarly, X1 is divided into quadrants, (i.e., X1→(reduced approximated image X2)+(edge extracted image in vertical direction)+(edge extracted image in horizontal direction)+(edge extracted image in diagonal direction), and then X2 is further divided into quadrants . . . , and up to XK is divided into quadrants. Repeating the division like this generates multi-divided images. This is a fundamental step of an image compression [1]. After the multi-division of the image, this image is coded by using SPIHT [5] and the like. Even a discrete cosine transformation requires the placing image X into blocks. However, in the case thereof, a division for an image into quadrants, as performed in the case of the block singular value decomposition, is not performed in the discrete cosine transformation. Quantization and coding by using the Huffman code are performed on the discrete cosine transformation, and a compressed image having a small amount of data and easy to enlarge and reduce a size of the compressed data is obtained. However, it should be noted that an original image of this compressed data cannot be restored and thus the discrete cosine transformation is an irreversible transformation [2].
An algorithm for multi-division by the block singular value decomposition is proposed by Kakarala-Ogunbona [3]. A hybrid-type algorithm using both this algorithm and the discrete wavelet transformation is also reported [1]. It is known that an image having special property such as a fingerprint is image-compressed by the hybrid-type algorithm so as to have more natural image quality than the discrete cosine transformation or the discrete wavelet transformation, which were put into practice as JEPG and JPEG 2000. An image compression performed by the block singular value decomposition includes various potentials, thus requiring a further numerical verification.    Non-patent Document Reference 1: [1] Ashino, R., Morimoto, A., Nagase, M., and Vaillancourt, R.: Image compression with multi-division singular value decomposition and other methods, Math. Comput. Model., Vol. 41, pp. 773. 790 (2005)    Non-patent Document Reference 2: [2] KOSHI Tomohiro, KURODA Hideo: JPEG & MPEG “Image compression technique understand able with illustration”, Nippon Jitsugyo Publishing (2006)    Non-patent Document Reference 3: [3] Kakarala, R., and Ogunbona, O. P.: Signal analysis using a multi-division form of the singular value decomposition, IEEE Trans. Image Process., Vol. 10, pp. 724. 735 (2001)    Non-patent Document Reference 4: [4] Iwasaki, M., and Nakamura, Y.: Accurate computation of singular values in terms of shifted integrable schemes, (submitted)    Non-patent Document Reference 5: [5] Said, A., and Pearlman, A. W.: A new fast and efficient image codec based on set partitioning in hierarchical trees, IEEE Trans. on Circuits and Systems for Video Technology, Vol. 6, pp. 243. 250 (1996)    Non-patent Document Reference 6: [6] TAKATA Masami, KIMURA Yasushi, IWASAKI Masashi, and NAKAMURA Yoshimasa: Library development for high speed singular value decomposition (submitted)    Non-patent Document Reference 7: [7] Parlett, B. N., and Marques, O. A.: An implementation of the dqds algorithm (positive case), Lin. Alg. Appl., Vol. 309, pp 217.259 (2000)